There are a number of sites like the one below that
appear to confirm the presence of “lists” maintained by the
FBI, by which persons seeking to purchase tickets and/ or
board planes, are “screened”; this work is done for various
uses including TSA use. If the uses a “list” based on
FBI “background checks”, their claim is ridiculous IMHO.

The FBI does background checks, prepares a “list” of various
checks, and makes that available to TSA. Both agencies are
large, powerful bureaucratic organizations with very significant
resources to check on travelers. It would also makes sense
that they would not divulge all the depth of details of what
they do, but I think the sources I’ve sent backs up the comment.